Math 5285H: Fundamental Structures of Algebra I (Fall 2011)

Lectures: MWF 10:10-11:00 in Vincent Hall 20.

Instructor: Gregg Musiker (musiker "at" math.umn.edu)

Office Hours: MWF 11:00-12:05 in Vincent 251; also by appointment.

Course Description:

This is the first semester of a course in the basic algebra of groups, ring, fields, and vector spaces. Roughly speaking, the Fall and Spring semesters should divide the topics as follows:

Fall) Vector spaces, linear algebra, group theory, symmetry, and the Sylow Theorems

Spring) Rings, fields, and Galois theory

Prerequisites: Some previous exposure to linear algebra (vectors, matrices, determinants) (such as from 2243, 2373, or 2573) would help. Also, one should either have the ability to write and read mathematical proofs (such as from 2283, 2574, or 3283), or have the desire and drive to learn how.

Required text: Algera, by Michael Artin, (2nd edition 2011, Prentice Hall).

Fall: some (but not all) of Chapters 1-7;

Spring: some (but not all) of Chapters 11,12,13,15,16

Note that I will be following the brand new second edition in this course, and encourage you to do the same. If you have the first edition, your text will more or less contain the same topics (although with occasional exceptions and sometimes in a different order), but the references to section numbers and exercises will not match the ones I announce in class and on the website. If you are in this situation, please find a classmate with the new edition or come talk to me early on.

Other useful texts (On reserve in math library)
Title Author(s) Year
Introduction to abstract algebra W. Keith Nicholson 2007
Contemporary abstract algebra Joseph A. Gallian 1994
Algebra Thomas Hungerford 1980
Abstract Algebra Dummit and Foote 2004
Abstract algebra: theory and applications Thomas Judson 1994

Grading:

  • Homework (50%): There will be 5 homework assignments due approximately every other week (tentatively) on Wednesdays in class. The first homework assignment is due on September 21st.

    I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates their collaborators. Late homework will not be accepted. Early homework is fine, and can be left in my mailbox in the School of Math mailroom in Vincent Hall 107. Homework solutions should be well-explained; the grader is told not to give credit for an unsupported answer. Complaints about the grading should be brought to me.

  • Exams (15% each): There will be 2 take-home exams, handed out on October 5th (due October 12th) and November 9th (due November 16th). Each will be open book, open library, open notes, and with calculators allowed. However, for these exams, you are not allowed to consult other electronic sources, such as the internet, and you are not allowed to collaborate or consult with other students or other human sources. These exams are to be collected in class.

  • Final Exam (20%): The final exam will be take-home as well, under the same policies as above, handed out on December 7th, to be turned in during class on December 14th.

  • Class Participation:

    Participation in class is encouraged. Please feel free to stop me and ask questions during lecture. Otherwise, I might stop and ask you questions instead.

    Course Syllabus and Tentative Lecture Schedule

  • (Sep 5) Lecture 1: Introduction to the Course
  • (Sep 7) Lecture 2: The Basic Matrix Operations (Artin, Sec. 1.1)
  • (Sep 12) Lecture 3: Row Reductions (Artin, Sec. 1.2)
  • (Sep 14) Lecture 4: Uniqueness of the Determinant (Artin, Sec. 1.3-1.4)
  • (Sep 16) Lecture 5: Permutations and other formulas for the Determinant (Artin, Sec. 1.5-1.6)
  • (Sep 19) Lecture 6: Introduction to Groups (Artin, Sec. 2.1)
  • (Sep 21) Lecture 7: Subgroups (Artin, Sec. 2.2-2.3)
  • (Sep 23) Lecture 8: Cyclic groups (Artin, Sec. 2.4)
  • (Sep 26) Lecture 9: Homomorphisms (Artin, Sec. 2.5)
  • (Sep 28) Lecture 10: Isomorphisms (Artin, Sec. 2.6)
  • (Sep 30) Lecture 11: Cosets and Modular Arithmetic (Artin, Sec. 2.7-2.9)
  • (Oct 3) Lecture 12: Normal subgroups and right cosets (Artin, Sec. 2.8)
  • (Oct 5) Lecture 13: Product groups and Quotient groups (Artin, Sec. 2.11-2.12)
  • (Oct 7) Lecture 14: Subspaces of R^n and Fields (Artin, Sec. 3.1-3.2)
  • (Oct 10) Lecture 15: Vector Spaces (Artin, Sec. 3.3)
  • (Oct 12) Lecture 16: More on Vector Spaces and Fields (Artin, Sec 3.3) (Take-Home Exam 1 due - Chapters 1-2)
  • (Oct 14) Lecture 17: Bases and Dimension (Artin, Sec. 3.4)
  • (Oct 17) Lecture 18: Computing with Bases (Artin, Sec. 3.5)
  • (Oct 19) Lecture 19: Direct Sums and Infinite Dimensional Spaces (Artin, Sec. 3.6-3.7)
  • (Oct 21) Lecture 20: The Dimension Formula and the Matrix of a Linear Transformation, Linear Operators (Artin, Sec. 4.1-4.2, glimpse of 4.3 and 4.7)
  • (Oct 24) Lecture 21: Orthogonal Matrices (Artin, Sec. 5.1)
  • (Oct 26) Lecture 22: SO3 and a quick review of Eigenvalues and Eigenvectors (Artin, Sec 4.4-4.5)
  • (Oct 28) Lecture 23: Symmetries and Isometries (Artin, Sec. 6.1-6.2)
  • (Oct 31) Lecture 24: Isometries of the Plane (Artin, Sec. 6.3)
  • (Nov 2) Lecture 25: Isometries of the Plane (cont.)
  • (Nov 4) Lecture 26: Finite and discrete groups of orthogonal operators on the plane (Artin, Sec. 6.4-6.5)
  • (Nov 7) Lecture 27: Discrete groups of isometries (Artin, Sec. 6.5-6.6)
  • (Nov 9) Lecture 28: Abstract symmetry and the operation on cosets (Artin, Sec. 6.7-6.8)
  • (Nov 11) Lecture 29: The counting formula and applications to the cube (Artin, Sec. 6.9-6.10)
  • (Nov 14) Lecture 30: Permutation Representations (Artin, Sec. 6.11)
  • (Nov 16) Lecture 31: Finite subgroups of S03 (Artin, Sec. 6.12) (Take-Home Exam 2 due - Sections 3.1-3.6, 4.1-4.5, 5.1, 6.1-6.5, 6.7-6.11)
  • (Nov 18) Lecture 32: Cayley's Theorem and the Class Equation (Artin, Sec. 7.1-7.2)
  • (Nov 21) Lecture 33: Class Equation and P-groups (Artin, Sec. 7.2-7.3)
  • (Nov 23) Lecture 34: The Class Equation of the Icosahedral group (Artin, Sec. 7.3-7.4)
  • (Nov 25) Thanksgiving Holiday:
  • (Nov 28) Lecture 35: Simple Groups (Artin, Sec. 7.5)
  • (Nov 30) Lecture 36: The Sylow Theorems: Part I (Artin, Sec. 7.6-7.7)
  • (Dec 2) Lecture 37: Application of the Sylow Theorems to Groups of Order 12 (Artin, Sec. 7.8)
  • (Dec 5) Lecture 38: The Sylow Theorems: Part II (proofs) (Artin, Sec. 7.7)
  • (Dec 7) Lecture 39: Towards a glimpse of Lie Groups and Cayley-Hamilton (Artin, Sec 5.2)
  • (Dec 9) Lecture 40: Bilinear Forms and a glimpse of the Classical Groups (Artin, Chapter 8 and Sec 9.1)
  • (Dec 12) Lecture 41: The Special Unitary Group SU2 (Artin, Sec. 9.2-9.4)
  • (Dec 14) Lecture 42: Preview of the Spring and Take-Home Final Due

  • Announcements

  • Office hours on Fridays now 1:25-2:30 instead of 11:00-12:05.
  • Take Home Exam 1 (due Oct 12th in class) is now posted.
  • Please see note below about numbering for problems in HW3.
  • Notes (post-grading) added to Solutions to Take Home Exam 1. Average was 84/100.
  • Errata for HW 4, please see below.
  • Take Home Exam 2 (due Nov 16th in class) is now posted.
    Homework assignments and exams
    Assignment or Exam Due date Problems from Artin text,
    unless otherwise specified
    Homework 1 Wednesday 9/21 Chapter 1: # 1.4, 1.6, 1.7, 1.10, 1.11, 1.13, 2.2, 2.5, 2.6, 2.10, 3.1, 4.1, 4.3, 5.1, 5.3, 5.4, 6.2, M.3, M.7
    Homework 2 Wednesday 10/5 Chapter 2: # 1.3, 2.1, 2.2, 2.4, 3.1, 3.2, 4.1, 4.3, 4.6, 4.7, 4.8, 4.9, 5.2, 5.3, 5.6, 6.2, 6.3, 6.6, 6.7, 6.10
    Exam 1 Wednesday 10/12 Download Here
    Homework 3 Wednesday 10/26 Chapter 3: # 1.2, 1.5, 1.6, 1.8, 1.9, 1.10, 1.11, 3.1, 3.2, 4.3, 4.4, 5.1, 5.2
    Chapter 4: # 1.3, 1.4, 2.3
    IMPORTANT NOTE: Some books number the exercises differently. In Chapter 3, Problems 1.2, ..., 1.11 should be in the section labeled "Fields", Problems 3.1, 3.2 should be in the section labeled "Bases and Dimension", Problems 4.3, 4.4 should be in the section labeled "Computing with Bases", Problems 5.1, 5.2 should be in the section labeled "Direct Sums".
    Homework 4 Wednesday 11/9 Chapter 4: # 3.2, 4.1, 4.2, 6.8
    Chapter 5: # 1.1, 1.3, M.1 (for n=2 and 3), M.5
    Typo in M.5: formula should be 1/\alpha ( [(1+\alpha)/2]^n - [(1-\alpha)/2]^n)
    Chapter 6: (Section 3 Isometries of the Plane): # 3.2, 3.3, 3.4,
    (Section 4 Finite Groups of Orthogonal Operators on the Plane): # 4.1, 4.2, 4.3,
    (Section 5 Discrete Groups of Isometries): # 5.1,
    (Section 6 Plane Crystallographic Groups): # 6.2
    Exam 2 Wednesday 11/16 Download Here
    Homework 5 Wednesday 12/7 Chapter 6: # 7.1, 7.3, 8.2, 10.2, 11.1, 12.4, 12.5
    Chapter 7: # 2.1, 2.2, 2.3, 2.7, 2.8, 2.9, 2.14, 2.17, 3.2, 3.3, 3.4, 4.4, 4.5
    Final Exam Wednesday 12/14 Download Here